'On the spot' refers to time, rather than location. If do a repair, for example, 'on the spot', it means immediately. 'On-site' is a term used work contexts to mean in that something happens, or is located, at the site in question rather than at another site. For example, a large production facility may have an 'on-site' maintenance team, whereas a smaller one may have to call in maintenance personnel from elsewhere if needed.